Asymptotics of Kolmogorov Diameters for Some Classes of Harmonic Functions on Spheroids

AbstractLet ΓKD be the unit ball of the space of all bounded harmonic functions in a domain D in R3, considered as a compact subset of the Banach space C(K), where K is a compact subset of D. The old problem about the exact asymptotics for Kolmogorov diameters (widths) of this set,lndk(ΓKD)∼−τk1/2,k→∞, is solved positively in the case when K and D are closed and open confocal spheroids, respectively (i.e., prolate or oblate ellipsoids of revolution). Using some special asymptotic formulas for the associated Legendre functions Pmn(coshσ) as n→∞ and m/n→γ∈[0, 1] (considered earlier by the second author), we show that the constant τ is some averaged characteristic of the pair of spheroids, expressed by means of a certain function of the variable γ, which appears within those asymptotics. Unlike the corresponding problem for analytic functions, quite well investigated, the harmonic functions case has been studied, up to now, only in the case of concentric balls